Set and logical operations

Set and logical operations

set

definition

We figurative and abstract things, symbols, called objects , a whole must be called by the objects that make up the collection , each composed of a collection of objects called elements .

In general, we do not contain any element of the set is called the empty set , denoted by \ (\ emptySet \) .

Containing a set of finite elements is called a finite set , comprising a set of elements called infinite infinite set .

nature

• 1. VARIATION : For a given set, which elements must vary.

• 2. Uncertainty : elements of the collection must be determined.

  - For example, "China's municipalities," constitute a collection, "our small river" does not constitute a collection.

  Common episodes

Nonnegative integers (natural numbers): \ (\ N \)

Positive integers: \ (\ N ^ * \) or \ (\ N _ + \)

Set of integers: \ (\ the Z-\)

Set of rational numbers: \ (\ Q \)

Real numbers: \ (\ R & lt \)

Show

We typically use a capital English letter \ (A, B, C, \ cdots \) represents a set, lowercase English letters \ (a, b, c, \ cdots \) represents an element in the collection.

If \ (A \) is the set \ (A \) element, say \ (A \) belonging to a set of \ (A \) , denoted \ (A \ in A \) ; Accordingly, if the if the \ (A \) B is not set \ (a \) element, say \ (a \) does not belong to the set of \ (a \) , denoted \ (a \ notin a \) .

Representation collection:

1. enumeration method : the elements enumerated, using "{}" enclose

EG. "Equation \ (x ^ 2 + x- 2 = 0 \) all real number of the" set enumeration method using a composition represented as \ (\ {1, -2 \} \) .

2. Description Method : In general, if the set \ (the I \) belonging to a set of \ (A \) of any one element \ (X \) has a characteristic \ (P (X) \) , not belonging to the set \ ( a \) no element having a characteristic \ (p (x) \) , called \ (p (x) \) is set \ (a \) characteristic properties. At this time, a set of \ (A \) may be described as \ (\ {x \ in I | p (x) \} \)

eg. disposed equation \ (x ^ 2 + x- 2 = 0 \) of the real roots \ (X \) , then the "equation \ (x ^ 2 + x- 2 = 0 \) all real roots" consisting of set \ (B \) represented by the described method was \ (\ {X \ in \ R & lt | X ^ 2 + X-2 = 0 \} \) , where \ (x ^ 2 + x- 2 = 0 \) is set \ (B \) characteristic properties.

The relationship between the set

Koshuwa Mako Collection

In general, for two sets \ (A, B \) , if the set \ (A \) of any one element is \ (B \) elements, said that the two sets have included relations, said the collection \ ( a \) is a set of \ (B \) of the subset , denoted by: \ (a \ B subseteq \) (or \ (B \ a supseteq \) ). Therefore, any one set \ (A \) are a subset of its own, i.e. \ (A \ A subseteq \) .

We require: the empty set is a subset of a set of arbitrary , that is, a collection for any \ (A \) , there \ (\ emptySet \ subseteq A \) .

If the set \ (A \) is a set of \ (B \) subset, and \ (B \) there is at least one element not belonging to \ (A \) , then the set \ (A \) is called a set of \ (B \ ) the subset , denoted by: \ (a \ B subsetneq \) or \ (B \ a supsetneq \) .

Using Venn map may be expressed as:

It can be inferred:

1. For the set \ (A, B, C \) , if the \ (A \ B subseteq, B \ subseteq C \) , then \ (A \ subseteq C \) .

2. For a set of \ (A, B, C \) , if the \ (A \ B subsetneq, B \ subsetneq C \) , then \ (A \ subsetneq C \) .

Power set

For a given set \ (A \) , set as a subset of all its elements called \ (A \) of the power set, denoted \ (P (A) \) .

If \ (| A | = n-\) , then \ (| P (A) | = 2 ^ {| A |} \) . (Where \ (| A | \) denote the set \ (A \) the number of elements, also known as a collection of \ (A \) potential)

Equal to the set

Generally, if the set \ (A \) for each element is set \ (B \) elements, and a set of \ (B \) for each element is set \ (A \) elements, then He said set \ (A \) equal to the set \ (B \) , denoted \ (A = B \) .

Available equivalent defined by: If \ (A \ B subseteq, B \ subseteq A \) , then the \ (A = B \) .

Collection operation

Intersection

Given by the set of two \ (A, B \) set of common elements constituting the set is called \ (A, B \) of intersection, referred to as \ (A \ CAP B \) .

FIG using Venn can be expressed as:

Defined by the intersection obtained, for any two sets \ (A, B \) , are:

\(A\cap B=B\cap A\)

\ (A \ cap A = A \)

\(A\cap \emptyset=\emptyset\cap A=\emptyset\)

If \ (A \ subseteq B, \ ) then \ (A \ cap B = A \)

Union

Given by the set of two \ (A, B \) the set of all elements constituting called \ (A, B \) and set, referred to as \ (A \ B Cup \) .

A Venn FIG be expressed as:

Defined by the union have, for any two sets \ (A, B \) , are:

\(A\cup B=B\cup A\)

\(A\cup A=A\)

\(A\cup \emptyset =\emptyset \cup A=A\)

If \ (A \ subseteq B, \ ) then \ (A \ cup B = B \)

Complement

In studying the relationship between the collection, a subset of the set if you want to study is a given set, this collection is called The Complete Works , usually \ (U \) represented.

Complete Works is relative.

For example, research number set, the set of real numbers constant \ (\ R & lt \) as the corpus; study when only a natural number, put the set of natural numbers \ (\ N \) as a corpus.

如果知道集合\(A\)是全集\(U\)的一个子集，由\(U\)中所有不属于\(A\)的元素组成的集合，叫做\(A\)在\(U\)中的(绝对)补集，记作：

\(\complement_UA\)(~\(A\))

用Venn图可表示为：

若给定集合\(A,B\)，则\(A\)在\(B\)中的相对补集(差集)由属于\(B\)而不属于\(A\)的元素组成，记作\(B-A\)。

\(A\)在\(B\)中的差集其实就可以理解为\(A\cap B\)在\(B\)中的补集。

对称差

对于给定的集合\(A,B\)，只属于其中一个集合，而不属于另一个集合的元素构成的集合叫做这个集合的对称差，记作：

\(A\Delta B\)(\(A\oplus B\))。

用Venn图可表示为：

集合运算在位运算中的表示

交集：A&B

并集：A|B

补集：~A

对称差：A^B

逻辑运算

逻辑连结词

在逻辑或数学中，我们常用逻辑词连结两个命题组成一个新命题，而常用的逻辑连结词有“且” “或” “非”。

且

若有命题\(p,q\)，用"且"连结可组成一个新命题，记作\(p\wedge q\)。只有\(p,q\)都是真命题时，\(p\wedge q\)才是真命题。

或

若有命题\(p,q\)，用"或"连结可组成一个新命题，记作\(p\vee q\)。只要\(p,q\)中有至少一个是真命题时，\(p\vee q\)就是真命题。

非

若有命题\(p\)，对它加以否定则构成一个新命题，记作\(\neg p\)。\(p\)和\(\neg p\)的真假相反。

异或的表示：\(a \oplus b=(\neg a\wedge b)\vee (a\wedge \neg b)\)

优先级：非>与>异或>或

量词

对于含有变量的语句，当对它赋予一个值或条件时就可以成为一个命题。

全称量词

表示陈述事物全体的量词叫做全称量词，用符号\(\forall\)表示；含有全称量词的命题，叫做全称命题。

设\(p(x)\)是某集合\(M\)所有元素都含有的性质，则一个全称命题可记为：\(\forall x\in M,p(x).\)

存在量词

表示陈述事物个体或部分的量词叫做存在量词，用符号\(\exist\)表示；含有存在量词的命题，叫做存在性命题。

类似的，一个存在性命题可记为\(\exists x\in M,p(x).\)